LMFDB - Embedded newform 1336.2.a.d.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} + \cdots + 64 $ x^12 - x^11 - 25x^10 + 20x^9 + 224x^8 - 135x^7 - 865x^6 + 330x^5 + 1341x^4 - 127x^3 - 652x^2 - 48x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.42555$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.42555 q^{3} -0.247339 q^{5} +2.30877 q^{7} +2.88329 q^{9} +O(q^{10})$	$q+2.42555 q^{3} -0.247339 q^{5} +2.30877 q^{7} +2.88329 q^{9} +0.611439 q^{11} -3.57276 q^{13} -0.599934 q^{15} +2.36266 q^{17} +2.58875 q^{19} +5.60004 q^{21} +6.55175 q^{23} -4.93882 q^{25} -0.283087 q^{27} +7.68437 q^{29} +9.82801 q^{31} +1.48308 q^{33} -0.571050 q^{35} +6.31725 q^{37} -8.66590 q^{39} -9.78170 q^{41} +0.757915 q^{43} -0.713151 q^{45} -9.70767 q^{47} -1.66957 q^{49} +5.73074 q^{51} -3.18874 q^{53} -0.151233 q^{55} +6.27914 q^{57} +3.12492 q^{59} -5.52747 q^{61} +6.65686 q^{63} +0.883683 q^{65} +3.92535 q^{67} +15.8916 q^{69} +4.50464 q^{71} +3.57893 q^{73} -11.9794 q^{75} +1.41167 q^{77} -3.75089 q^{79} -9.33651 q^{81} -8.82555 q^{83} -0.584378 q^{85} +18.6388 q^{87} -1.73784 q^{89} -8.24868 q^{91} +23.8383 q^{93} -0.640299 q^{95} -18.8467 q^{97} +1.76296 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.42555	1.40039	0.700196	−	0.713951i	$-0.253096\pi$
$3$	2.42555	1.40039	0.700196	+	0.713951i	$0.253096\pi$
$4$	0	0
$5$	−0.247339	−0.110614	−0.0553068	−	0.998469i	$-0.517614\pi$
$5$	−0.247339	−0.110614	−0.0553068	+	0.998469i	$0.517614\pi$
$6$	0	0
$7$	2.30877	0.872634	0.436317	−	0.899793i	$-0.356283\pi$
$7$	2.30877	0.872634	0.436317	+	0.899793i	$0.356283\pi$
$8$	0	0
$9$	2.88329	0.961097
$10$	0	0
$11$	0.611439	0.184356	0.0921779	−	0.995743i	$-0.470617\pi$
$11$	0.611439	0.184356	0.0921779	+	0.995743i	$0.470617\pi$
$12$	0	0
$13$	−3.57276	−0.990904	−0.495452	−	0.868635i	$-0.664998\pi$
$13$	−3.57276	−0.990904	−0.495452	+	0.868635i	$0.664998\pi$
$14$	0	0
$15$	−0.599934	−0.154902
$16$	0	0
$17$	2.36266	0.573028	0.286514	−	0.958076i	$-0.407503\pi$
$17$	2.36266	0.573028	0.286514	+	0.958076i	$0.407503\pi$
$18$	0	0
$19$	2.58875	0.593900	0.296950	−	0.954893i	$-0.404031\pi$
$19$	2.58875	0.593900	0.296950	+	0.954893i	$0.404031\pi$
$20$	0	0
$21$	5.60004	1.22203
$22$	0	0
$23$	6.55175	1.36613	0.683067	−	0.730355i	$-0.260646\pi$
$23$	6.55175	1.36613	0.683067	+	0.730355i	$0.260646\pi$
$24$	0	0
$25$	−4.93882	−0.987765
$26$	0	0
$27$	−0.283087	−0.0544801
$28$	0	0
$29$	7.68437	1.42695	0.713475	−	0.700680i	$-0.247120\pi$
$29$	7.68437	1.42695	0.713475	+	0.700680i	$0.247120\pi$
$30$	0	0
$31$	9.82801	1.76516	0.882581	−	0.470161i	$-0.155804\pi$
$31$	9.82801	1.76516	0.882581	+	0.470161i	$0.155804\pi$
$32$	0	0
$33$	1.48308	0.258170
$34$	0	0
$35$	−0.571050	−0.0965251
$36$	0	0
$37$	6.31725	1.03855	0.519275	−	0.854607i	$-0.326202\pi$
$37$	6.31725	1.03855	0.519275	+	0.854607i	$0.326202\pi$
$38$	0	0
$39$	−8.66590	−1.38765
$40$	0	0
$41$	−9.78170	−1.52765	−0.763823	−	0.645426i	$-0.776680\pi$
$41$	−9.78170	−1.52765	−0.763823	+	0.645426i	$0.776680\pi$
$42$	0	0
$43$	0.757915	0.115581	0.0577905	−	0.998329i	$-0.481594\pi$
$43$	0.757915	0.115581	0.0577905	+	0.998329i	$0.481594\pi$
$44$	0	0
$45$	−0.713151	−0.106310
$46$	0	0
$47$	−9.70767	−1.41601	−0.708004	−	0.706208i	$-0.750404\pi$
$47$	−9.70767	−1.41601	−0.708004	+	0.706208i	$0.750404\pi$
$48$	0	0
$49$	−1.66957	−0.238510
$50$	0	0
$51$	5.73074	0.802464
$52$	0	0
$53$	−3.18874	−0.438007	−0.219003	−	0.975724i	$-0.570281\pi$
$53$	−3.18874	−0.438007	−0.219003	+	0.975724i	$0.570281\pi$
$54$	0	0
$55$	−0.151233	−0.0203922
$56$	0	0
$57$	6.27914	0.831692
$58$	0	0
$59$	3.12492	0.406830	0.203415	−	0.979093i	$-0.434796\pi$
$59$	3.12492	0.406830	0.203415	+	0.979093i	$0.434796\pi$
$60$	0	0
$61$	−5.52747	−0.707720	−0.353860	−	0.935298i	$-0.615131\pi$
$61$	−5.52747	−0.707720	−0.353860	+	0.935298i	$0.615131\pi$
$62$	0	0
$63$	6.65686	0.838685
$64$	0	0
$65$	0.883683	0.109607
$66$	0	0
$67$	3.92535	0.479558	0.239779	−	0.970828i	$-0.422925\pi$
$67$	3.92535	0.479558	0.239779	+	0.970828i	$0.422925\pi$
$68$	0	0
$69$	15.8916	1.91312
$70$	0	0
$71$	4.50464	0.534603	0.267301	−	0.963613i	$-0.413868\pi$
$71$	4.50464	0.534603	0.267301	+	0.963613i	$0.413868\pi$
$72$	0	0
$73$	3.57893	0.418882	0.209441	−	0.977821i	$-0.432836\pi$
$73$	3.57893	0.418882	0.209441	+	0.977821i	$0.432836\pi$
$74$	0	0
$75$	−11.9794	−1.38326
$76$	0	0
$77$	1.41167	0.160875
$78$	0	0
$79$	−3.75089	−0.422008	−0.211004	−	0.977485i	$-0.567673\pi$
$79$	−3.75089	−0.422008	−0.211004	+	0.977485i	$0.567673\pi$
$80$	0	0
$81$	−9.33651	−1.03739
$82$	0	0
$83$	−8.82555	−0.968730	−0.484365	−	0.874866i	$-0.660949\pi$
$83$	−8.82555	−0.968730	−0.484365	+	0.874866i	$0.660949\pi$
$84$	0	0
$85$	−0.584378	−0.0633847
$86$	0	0
$87$	18.6388	1.99829
$88$	0	0
$89$	−1.73784	−0.184210	−0.0921051	−	0.995749i	$-0.529360\pi$
$89$	−1.73784	−0.184210	−0.0921051	+	0.995749i	$0.529360\pi$
$90$	0	0
$91$	−8.24868	−0.864697
$92$	0	0
$93$	23.8383	2.47192
$94$	0	0
$95$	−0.640299	−0.0656933
$96$	0	0
$97$	−18.8467	−1.91359	−0.956796	−	0.290759i	$-0.906092\pi$
$97$	−18.8467	−1.91359	−0.956796	+	0.290759i	$0.906092\pi$
$98$	0	0
$99$	1.76296	0.177184
$100$	0	0
$101$	17.9147	1.78258	0.891289	−	0.453435i	$-0.149802\pi$
$101$	17.9147	1.78258	0.891289	+	0.453435i	$0.149802\pi$
$102$	0	0
$103$	−18.6184	−1.83452	−0.917261	−	0.398288i	$-0.869605\pi$
$103$	−18.6184	−1.83452	−0.917261	+	0.398288i	$0.869605\pi$
$104$	0	0
$105$	−1.38511	−0.135173
$106$	0	0
$107$	−7.82319	−0.756296	−0.378148	−	0.925745i	$-0.623439\pi$
$107$	−7.82319	−0.756296	−0.378148	+	0.925745i	$0.623439\pi$
$108$	0	0
$109$	14.9726	1.43411	0.717056	−	0.697015i	$-0.245489\pi$
$109$	14.9726	1.43411	0.717056	+	0.697015i	$0.245489\pi$
$110$	0	0
$111$	15.3228	1.45438
$112$	0	0
$113$	15.1238	1.42273	0.711364	−	0.702824i	$-0.248078\pi$
$113$	15.1238	1.42273	0.711364	+	0.702824i	$0.248078\pi$
$114$	0	0
$115$	−1.62051	−0.151113
$116$	0	0
$117$	−10.3013	−0.952355
$118$	0	0
$119$	5.45483	0.500044
$120$	0	0
$121$	−10.6261	−0.966013
$122$	0	0
$123$	−23.7260	−2.13930
$124$	0	0
$125$	2.45826	0.219874
$126$	0	0
$127$	−6.01366	−0.533626	−0.266813	−	0.963748i	$-0.585971\pi$
$127$	−6.01366	−0.533626	−0.266813	+	0.963748i	$0.585971\pi$
$128$	0	0
$129$	1.83836	0.161859
$130$	0	0
$131$	4.55174	0.397687	0.198843	−	0.980031i	$-0.436281\pi$
$131$	4.55174	0.397687	0.198843	+	0.980031i	$0.436281\pi$
$132$	0	0
$133$	5.97683	0.518257
$134$	0	0
$135$	0.0700185	0.00602623
$136$	0	0
$137$	11.1161	0.949712	0.474856	−	0.880063i	$-0.342500\pi$
$137$	11.1161	0.949712	0.474856	+	0.880063i	$0.342500\pi$
$138$	0	0
$139$	−8.75102	−0.742251	−0.371126	−	0.928583i	$-0.621028\pi$
$139$	−8.75102	−0.742251	−0.371126	+	0.928583i	$0.621028\pi$
$140$	0	0
$141$	−23.5464	−1.98297
$142$	0	0
$143$	−2.18452	−0.182679
$144$	0	0
$145$	−1.90065	−0.157840
$146$	0	0
$147$	−4.04963	−0.334008
$148$	0	0
$149$	−12.2247	−1.00148	−0.500742	−	0.865596i	$-0.666940\pi$
$149$	−12.2247	−1.00148	−0.500742	+	0.865596i	$0.666940\pi$
$150$	0	0
$151$	−1.62956	−0.132612	−0.0663058	−	0.997799i	$-0.521121\pi$
$151$	−1.62956	−0.132612	−0.0663058	+	0.997799i	$0.521121\pi$
$152$	0	0
$153$	6.81222	0.550735
$154$	0	0
$155$	−2.43085	−0.195251
$156$	0	0
$157$	−19.6933	−1.57170	−0.785849	−	0.618419i	$-0.787774\pi$
$157$	−19.6933	−1.57170	−0.785849	+	0.618419i	$0.787774\pi$
$158$	0	0
$159$	−7.73444	−0.613381
$160$	0	0
$161$	15.1265	1.19214
$162$	0	0
$163$	−21.0616	−1.64967	−0.824834	−	0.565375i	$-0.808732\pi$
$163$	−21.0616	−1.64967	−0.824834	+	0.565375i	$0.808732\pi$
$164$	0	0
$165$	−0.366823	−0.0285571
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−0.235413	−0.0181087
$170$	0	0
$171$	7.46411	0.570795
$172$	0	0
$173$	−7.14346	−0.543107	−0.271554	−	0.962423i	$-0.587537\pi$
$173$	−7.14346	−0.543107	−0.271554	+	0.962423i	$0.587537\pi$
$174$	0	0
$175$	−11.4026	−0.861957
$176$	0	0
$177$	7.57964	0.569721
$178$	0	0
$179$	3.84457	0.287356	0.143678	−	0.989624i	$-0.454107\pi$
$179$	3.84457	0.287356	0.143678	+	0.989624i	$0.454107\pi$
$180$	0	0
$181$	9.86331	0.733134	0.366567	−	0.930392i	$-0.380533\pi$
$181$	9.86331	0.733134	0.366567	+	0.930392i	$0.380533\pi$
$182$	0	0
$183$	−13.4072	−0.991086
$184$	0	0
$185$	−1.56251	−0.114878
$186$	0	0
$187$	1.44462	0.105641
$188$	0	0
$189$	−0.653583	−0.0475411
$190$	0	0
$191$	−1.71978	−0.124439	−0.0622194	−	0.998062i	$-0.519818\pi$
$191$	−1.71978	−0.124439	−0.0622194	+	0.998062i	$0.519818\pi$
$192$	0	0
$193$	22.9920	1.65500	0.827500	−	0.561465i	$-0.189762\pi$
$193$	22.9920	1.65500	0.827500	+	0.561465i	$0.189762\pi$
$194$	0	0
$195$	2.14342	0.153493
$196$	0	0
$197$	22.0079	1.56800	0.784000	−	0.620761i	$-0.213176\pi$
$197$	22.0079	1.56800	0.784000	+	0.620761i	$0.213176\pi$
$198$	0	0
$199$	−8.11037	−0.574929	−0.287465	−	0.957791i	$-0.592812\pi$
$199$	−8.11037	−0.574929	−0.287465	+	0.957791i	$0.592812\pi$
$200$	0	0
$201$	9.52113	0.671569
$202$	0	0
$203$	17.7414	1.24521
$204$	0	0
$205$	2.41940	0.168978
$206$	0	0
$207$	18.8906	1.31299
$208$	0	0
$209$	1.58286	0.109489
$210$	0	0
$211$	−3.23564	−0.222750	−0.111375	−	0.993778i	$-0.535526\pi$
$211$	−3.23564	−0.222750	−0.111375	+	0.993778i	$0.535526\pi$
$212$	0	0
$213$	10.9262	0.748653
$214$	0	0
$215$	−0.187462	−0.0127848
$216$	0	0
$217$	22.6906	1.54034
$218$	0	0
$219$	8.68087	0.586599
$220$	0	0
$221$	−8.44119	−0.567816
$222$	0	0
$223$	−5.03217	−0.336979	−0.168490	−	0.985703i	$-0.553889\pi$
$223$	−5.03217	−0.336979	−0.168490	+	0.985703i	$0.553889\pi$
$224$	0	0
$225$	−14.2401	−0.949337
$226$	0	0
$227$	−1.89805	−0.125978	−0.0629891	−	0.998014i	$-0.520063\pi$
$227$	−1.89805	−0.125978	−0.0629891	+	0.998014i	$0.520063\pi$
$228$	0	0
$229$	13.3110	0.879615	0.439808	−	0.898092i	$-0.355047\pi$
$229$	13.3110	0.879615	0.439808	+	0.898092i	$0.355047\pi$
$230$	0	0
$231$	3.42408	0.225288
$232$	0	0
$233$	−9.93439	−0.650823	−0.325412	−	0.945572i	$-0.605503\pi$
$233$	−9.93439	−0.650823	−0.325412	+	0.945572i	$0.605503\pi$
$234$	0	0
$235$	2.40109	0.156630
$236$	0	0
$237$	−9.09797	−0.590976
$238$	0	0
$239$	−7.78654	−0.503670	−0.251835	−	0.967770i	$-0.581034\pi$
$239$	−7.78654	−0.503670	−0.251835	+	0.967770i	$0.581034\pi$
$240$	0	0
$241$	−18.1258	−1.16758	−0.583792	−	0.811903i	$-0.698432\pi$
$241$	−18.1258	−1.16758	−0.583792	+	0.811903i	$0.698432\pi$
$242$	0	0
$243$	−21.7969	−1.39827
$244$	0	0
$245$	0.412951	0.0263825
$246$	0	0
$247$	−9.24897	−0.588498
$248$	0	0
$249$	−21.4068	−1.35660
$250$	0	0
$251$	−3.89199	−0.245661	−0.122830	−	0.992428i	$-0.539197\pi$
$251$	−3.89199	−0.245661	−0.122830	+	0.992428i	$0.539197\pi$
$252$	0	0
$253$	4.00600	0.251855
$254$	0	0
$255$	−1.41744	−0.0887634
$256$	0	0
$257$	18.1646	1.13308	0.566538	−	0.824036i	$-0.308283\pi$
$257$	18.1646	1.13308	0.566538	+	0.824036i	$0.308283\pi$
$258$	0	0
$259$	14.5851	0.906274
$260$	0	0
$261$	22.1563	1.37144
$262$	0	0
$263$	13.2227	0.815348	0.407674	−	0.913128i	$-0.366340\pi$
$263$	13.2227	0.815348	0.407674	+	0.913128i	$0.366340\pi$
$264$	0	0
$265$	0.788701	0.0484495
$266$	0	0
$267$	−4.21521	−0.257966
$268$	0	0
$269$	2.49832	0.152325	0.0761625	−	0.997095i	$-0.475733\pi$
$269$	2.49832	0.152325	0.0761625	+	0.997095i	$0.475733\pi$
$270$	0	0
$271$	−15.4964	−0.941342	−0.470671	−	0.882309i	$-0.655988\pi$
$271$	−15.4964	−0.941342	−0.470671	+	0.882309i	$0.655988\pi$
$272$	0	0
$273$	−20.0076	−1.21091
$274$	0	0
$275$	−3.01979	−0.182100
$276$	0	0
$277$	8.42664	0.506308	0.253154	−	0.967426i	$-0.418532\pi$
$277$	8.42664	0.506308	0.253154	+	0.967426i	$0.418532\pi$
$278$	0	0
$279$	28.3370	1.69649
$280$	0	0
$281$	−5.05904	−0.301797	−0.150898	−	0.988549i	$-0.548217\pi$
$281$	−5.05904	−0.301797	−0.150898	+	0.988549i	$0.548217\pi$
$282$	0	0
$283$	19.4899	1.15855	0.579276	−	0.815131i	$-0.303335\pi$
$283$	19.4899	1.15855	0.579276	+	0.815131i	$0.303335\pi$
$284$	0	0
$285$	−1.55308	−0.0919964
$286$	0	0
$287$	−22.5837	−1.33308
$288$	0	0
$289$	−11.4179	−0.671639
$290$	0	0
$291$	−45.7136	−2.67978
$292$	0	0
$293$	−15.1744	−0.886497	−0.443248	−	0.896399i	$-0.646174\pi$
$293$	−15.1744	−0.886497	−0.443248	+	0.896399i	$0.646174\pi$
$294$	0	0
$295$	−0.772915	−0.0450009
$296$	0	0
$297$	−0.173090	−0.0100437
$298$	0	0
$299$	−23.4078	−1.35371
$300$	0	0
$301$	1.74985	0.100860
$302$	0	0
$303$	43.4530	2.49631
$304$	0	0
$305$	1.36716	0.0782835
$306$	0	0
$307$	−13.4861	−0.769693	−0.384847	−	0.922981i	$-0.625746\pi$
$307$	−13.4861	−0.769693	−0.384847	+	0.922981i	$0.625746\pi$
$308$	0	0
$309$	−45.1597	−2.56905
$310$	0	0
$311$	32.3582	1.83487	0.917433	−	0.397890i	$-0.130258\pi$
$311$	32.3582	1.83487	0.917433	+	0.397890i	$0.130258\pi$
$312$	0	0
$313$	7.21549	0.407844	0.203922	−	0.978987i	$-0.434631\pi$
$313$	7.21549	0.407844	0.203922	+	0.978987i	$0.434631\pi$
$314$	0	0
$315$	−1.64650	−0.0927700
$316$	0	0
$317$	−12.9027	−0.724685	−0.362343	−	0.932045i	$-0.618023\pi$
$317$	−12.9027	−0.724685	−0.362343	+	0.932045i	$0.618023\pi$
$318$	0	0
$319$	4.69852	0.263067
$320$	0	0
$321$	−18.9755	−1.05911
$322$	0	0
$323$	6.11632	0.340321
$324$	0	0
$325$	17.6452	0.978780
$326$	0	0
$327$	36.3167	2.00832
$328$	0	0
$329$	−22.4128	−1.23566
$330$	0	0
$331$	−11.3996	−0.626576	−0.313288	−	0.949658i	$-0.601431\pi$
$331$	−11.3996	−0.626576	−0.313288	+	0.949658i	$0.601431\pi$
$332$	0	0
$333$	18.2145	0.998147
$334$	0	0
$335$	−0.970894	−0.0530456
$336$	0	0
$337$	4.45701	0.242789	0.121394	−	0.992604i	$-0.461263\pi$
$337$	4.45701	0.242789	0.121394	+	0.992604i	$0.461263\pi$
$338$	0	0
$339$	36.6835	1.99238
$340$	0	0
$341$	6.00923	0.325418
$342$	0	0
$343$	−20.0161	−1.08077
$344$	0	0
$345$	−3.93062	−0.211617
$346$	0	0
$347$	−17.6634	−0.948222	−0.474111	−	0.880465i	$-0.657231\pi$
$347$	−17.6634	−0.948222	−0.474111	+	0.880465i	$0.657231\pi$
$348$	0	0
$349$	−23.6814	−1.26763	−0.633817	−	0.773483i	$-0.718513\pi$
$349$	−23.6814	−1.26763	−0.633817	+	0.773483i	$0.718513\pi$
$350$	0	0
$351$	1.01140	0.0539845
$352$	0	0
$353$	−12.2979	−0.654550	−0.327275	−	0.944929i	$-0.606130\pi$
$353$	−12.2979	−0.654550	−0.327275	+	0.944929i	$0.606130\pi$
$354$	0	0
$355$	−1.11418	−0.0591343
$356$	0	0
$357$	13.2310	0.700257
$358$	0	0
$359$	−21.8538	−1.15340	−0.576700	−	0.816956i	$-0.695660\pi$
$359$	−21.8538	−1.15340	−0.576700	+	0.816956i	$0.695660\pi$
$360$	0	0
$361$	−12.2984	−0.647283
$362$	0	0
$363$	−25.7742	−1.35280
$364$	0	0
$365$	−0.885210	−0.0463340
$366$	0	0
$367$	−30.2187	−1.57740	−0.788702	−	0.614775i	$-0.789247\pi$
$367$	−30.2187	−1.57740	−0.788702	+	0.614775i	$0.789247\pi$
$368$	0	0
$369$	−28.2035	−1.46821
$370$	0	0
$371$	−7.36207	−0.382220
$372$	0	0
$373$	22.8203	1.18159	0.590795	−	0.806822i	$-0.298814\pi$
$373$	22.8203	1.18159	0.590795	+	0.806822i	$0.298814\pi$
$374$	0	0
$375$	5.96264	0.307909
$376$	0	0
$377$	−27.4544	−1.41397
$378$	0	0
$379$	−9.74612	−0.500625	−0.250312	−	0.968165i	$-0.580533\pi$
$379$	−9.74612	−0.500625	−0.250312	+	0.968165i	$0.580533\pi$
$380$	0	0
$381$	−14.5864	−0.747286
$382$	0	0
$383$	−9.30510	−0.475468	−0.237734	−	0.971330i	$-0.576405\pi$
$383$	−9.30510	−0.475468	−0.237734	+	0.971330i	$0.576405\pi$
$384$	0	0
$385$	−0.349162	−0.0177950
$386$	0	0
$387$	2.18529	0.111085
$388$	0	0
$389$	9.78596	0.496168	0.248084	−	0.968739i	$-0.420199\pi$
$389$	9.78596	0.496168	0.248084	+	0.968739i	$0.420199\pi$
$390$	0	0
$391$	15.4795	0.782834
$392$	0	0
$393$	11.0405	0.556917
$394$	0	0
$395$	0.927743	0.0466798
$396$	0	0
$397$	−16.4532	−0.825762	−0.412881	−	0.910785i	$-0.635477\pi$
$397$	−16.4532	−0.825762	−0.412881	+	0.910785i	$0.635477\pi$
$398$	0	0
$399$	14.4971	0.725763
$400$	0	0
$401$	15.4919	0.773631	0.386815	−	0.922157i	$-0.373575\pi$
$401$	15.4919	0.773631	0.386815	+	0.922157i	$0.373575\pi$
$402$	0	0
$403$	−35.1131	−1.74911
$404$	0	0
$405$	2.30929	0.114749
$406$	0	0
$407$	3.86261	0.191463
$408$	0	0
$409$	−15.6560	−0.774142	−0.387071	−	0.922050i	$-0.626513\pi$
$409$	−15.6560	−0.774142	−0.387071	+	0.922050i	$0.626513\pi$
$410$	0	0
$411$	26.9626	1.32997
$412$	0	0
$413$	7.21472	0.355013
$414$	0	0
$415$	2.18291	0.107155
$416$	0	0
$417$	−21.2260	−1.03944
$418$	0	0
$419$	12.1824	0.595151	0.297576	−	0.954698i	$-0.403822\pi$
$419$	12.1824	0.595151	0.297576	+	0.954698i	$0.403822\pi$
$420$	0	0
$421$	−25.3297	−1.23450	−0.617248	−	0.786769i	$-0.711752\pi$
$421$	−25.3297	−1.23450	−0.617248	+	0.786769i	$0.711752\pi$
$422$	0	0
$423$	−27.9900	−1.36092
$424$	0	0
$425$	−11.6687	−0.566017
$426$	0	0
$427$	−12.7617	−0.617581
$428$	0	0
$429$	−5.29867	−0.255822
$430$	0	0
$431$	28.2494	1.36073	0.680363	−	0.732875i	$-0.261822\pi$
$431$	28.2494	1.36073	0.680363	+	0.732875i	$0.261822\pi$
$432$	0	0
$433$	0.832999	0.0400314	0.0200157	−	0.999800i	$-0.493628\pi$
$433$	0.832999	0.0400314	0.0200157	+	0.999800i	$0.493628\pi$
$434$	0	0
$435$	−4.61011	−0.221038
$436$	0	0
$437$	16.9608	0.811347
$438$	0	0
$439$	18.0075	0.859452	0.429726	−	0.902959i	$-0.358610\pi$
$439$	18.0075	0.859452	0.429726	+	0.902959i	$0.358610\pi$
$440$	0	0
$441$	−4.81386	−0.229231
$442$	0	0
$443$	32.5123	1.54471	0.772353	−	0.635194i	$-0.219080\pi$
$443$	32.5123	1.54471	0.772353	+	0.635194i	$0.219080\pi$
$444$	0	0
$445$	0.429835	0.0203761
$446$	0	0
$447$	−29.6516	−1.40247
$448$	0	0
$449$	−0.417893	−0.0197216	−0.00986081	−	0.999951i	$-0.503139\pi$
$449$	−0.417893	−0.0197216	−0.00986081	+	0.999951i	$0.503139\pi$
$450$	0	0
$451$	−5.98092	−0.281630
$452$	0	0
$453$	−3.95258	−0.185708
$454$	0	0
$455$	2.04022	0.0956472
$456$	0	0
$457$	4.34194	0.203107	0.101554	−	0.994830i	$-0.467619\pi$
$457$	4.34194	0.203107	0.101554	+	0.994830i	$0.467619\pi$
$458$	0	0
$459$	−0.668836	−0.0312186
$460$	0	0
$461$	38.1177	1.77532	0.887660	−	0.460499i	$-0.152330\pi$
$461$	38.1177	1.77532	0.887660	+	0.460499i	$0.152330\pi$
$462$	0	0
$463$	41.1465	1.91224	0.956121	−	0.292973i	$-0.0946447\pi$
$463$	41.1465	1.91224	0.956121	+	0.292973i	$0.0946447\pi$
$464$	0	0
$465$	−5.89615	−0.273428
$466$	0	0
$467$	15.6903	0.726060	0.363030	−	0.931777i	$-0.381742\pi$
$467$	15.6903	0.726060	0.363030	+	0.931777i	$0.381742\pi$
$468$	0	0
$469$	9.06274	0.418478
$470$	0	0
$471$	−47.7671	−2.20099
$472$	0	0
$473$	0.463419	0.0213080
$474$	0	0
$475$	−12.7854	−0.586633
$476$	0	0
$477$	−9.19406	−0.420967
$478$	0	0
$479$	−13.2874	−0.607117	−0.303559	−	0.952813i	$-0.598175\pi$
$479$	−13.2874	−0.607117	−0.303559	+	0.952813i	$0.598175\pi$
$480$	0	0
$481$	−22.5700	−1.02910
$482$	0	0
$483$	36.6901	1.66946
$484$	0	0
$485$	4.66153	0.211669
$486$	0	0
$487$	−24.3777	−1.10466	−0.552329	−	0.833626i	$-0.686261\pi$
$487$	−24.3777	−1.10466	−0.552329	+	0.833626i	$0.686261\pi$
$488$	0	0
$489$	−51.0858	−2.31018
$490$	0	0
$491$	14.5748	0.657753	0.328876	−	0.944373i	$-0.393330\pi$
$491$	14.5748	0.657753	0.328876	+	0.944373i	$0.393330\pi$
$492$	0	0
$493$	18.1555	0.817683
$494$	0	0
$495$	−0.436048	−0.0195989
$496$	0	0
$497$	10.4002	0.466512
$498$	0	0
$499$	16.7085	0.747976	0.373988	−	0.927434i	$-0.377990\pi$
$499$	16.7085	0.747976	0.373988	+	0.927434i	$0.377990\pi$
$500$	0	0
$501$	−2.42555	−0.108366
$502$	0	0
$503$	13.5395	0.603697	0.301849	−	0.953356i	$-0.402396\pi$
$503$	13.5395	0.603697	0.301849	+	0.953356i	$0.402396\pi$
$504$	0	0
$505$	−4.43101	−0.197177
$506$	0	0
$507$	−0.571005	−0.0253592
$508$	0	0
$509$	−15.0352	−0.666422	−0.333211	−	0.942852i	$-0.608132\pi$
$509$	−15.0352	−0.666422	−0.333211	+	0.942852i	$0.608132\pi$
$510$	0	0
$511$	8.26293	0.365531
$512$	0	0
$513$	−0.732840	−0.0323557
$514$	0	0
$515$	4.60505	0.202923
$516$	0	0
$517$	−5.93565	−0.261049
$518$	0	0
$519$	−17.3268	−0.760563
$520$	0	0
$521$	−26.8887	−1.17802	−0.589008	−	0.808127i	$-0.700481\pi$
$521$	−26.8887	−1.17802	−0.589008	+	0.808127i	$0.700481\pi$
$522$	0	0
$523$	13.3766	0.584917	0.292459	−	0.956278i	$-0.405527\pi$
$523$	13.3766	0.584917	0.292459	+	0.956278i	$0.405527\pi$
$524$	0	0
$525$	−27.6576	−1.20708
$526$	0	0
$527$	23.2202	1.01149
$528$	0	0
$529$	19.9254	0.866324
$530$	0	0
$531$	9.01004	0.391003
$532$	0	0
$533$	34.9476	1.51375
$534$	0	0
$535$	1.93498	0.0836566
$536$	0	0
$537$	9.32519	0.402412
$538$	0	0
$539$	−1.02084	−0.0439707
$540$	0	0
$541$	31.4398	1.35170	0.675852	−	0.737038i	$-0.263776\pi$
$541$	31.4398	1.35170	0.675852	+	0.737038i	$0.263776\pi$
$542$	0	0
$543$	23.9239	1.02667
$544$	0	0
$545$	−3.70331	−0.158632
$546$	0	0
$547$	−22.0417	−0.942436	−0.471218	−	0.882017i	$-0.656186\pi$
$547$	−22.0417	−0.942436	−0.471218	+	0.882017i	$0.656186\pi$
$548$	0	0
$549$	−15.9373	−0.680188
$550$	0	0
$551$	19.8929	0.847466
$552$	0	0
$553$	−8.65995	−0.368258
$554$	0	0
$555$	−3.78993	−0.160874
$556$	0	0
$557$	−12.5709	−0.532648	−0.266324	−	0.963884i	$-0.585809\pi$
$557$	−12.5709	−0.532648	−0.266324	+	0.963884i	$0.585809\pi$
$558$	0	0
$559$	−2.70785	−0.114530
$560$	0	0
$561$	3.50400	0.147939
$562$	0	0
$563$	12.6446	0.532907	0.266453	−	0.963848i	$-0.414148\pi$
$563$	12.6446	0.532907	0.266453	+	0.963848i	$0.414148\pi$
$564$	0	0
$565$	−3.74071	−0.157373
$566$	0	0
$567$	−21.5559	−0.905262
$568$	0	0
$569$	−36.1294	−1.51462	−0.757312	−	0.653053i	$-0.773488\pi$
$569$	−36.1294	−1.51462	−0.757312	+	0.653053i	$0.773488\pi$
$570$	0	0
$571$	24.6799	1.03282	0.516410	−	0.856341i	$-0.327268\pi$
$571$	24.6799	1.03282	0.516410	+	0.856341i	$0.327268\pi$
$572$	0	0
$573$	−4.17141	−0.174263
$574$	0	0
$575$	−32.3579	−1.34942
$576$	0	0
$577$	−25.3798	−1.05657	−0.528287	−	0.849066i	$-0.677166\pi$
$577$	−25.3798	−1.05657	−0.528287	+	0.849066i	$0.677166\pi$
$578$	0	0
$579$	55.7682	2.31765
$580$	0	0
$581$	−20.3762	−0.845346
$582$	0	0
$583$	−1.94972	−0.0807491
$584$	0	0
$585$	2.54792	0.105343
$586$	0	0
$587$	45.6589	1.88454	0.942272	−	0.334848i	$-0.108685\pi$
$587$	45.6589	1.88454	0.942272	+	0.334848i	$0.108685\pi$
$588$	0	0
$589$	25.4422	1.04833
$590$	0	0
$591$	53.3813	2.19581
$592$	0	0
$593$	26.5182	1.08897	0.544485	−	0.838770i	$-0.316725\pi$
$593$	26.5182	1.08897	0.544485	+	0.838770i	$0.316725\pi$
$594$	0	0
$595$	−1.34920	−0.0553116
$596$	0	0
$597$	−19.6721	−0.805126
$598$	0	0
$599$	2.40684	0.0983410	0.0491705	−	0.998790i	$-0.484342\pi$
$599$	2.40684	0.0983410	0.0491705	+	0.998790i	$0.484342\pi$
$600$	0	0
$601$	−15.9043	−0.648749	−0.324374	−	0.945929i	$-0.605154\pi$
$601$	−15.9043	−0.648749	−0.324374	+	0.945929i	$0.605154\pi$
$602$	0	0
$603$	11.3179	0.460901
$604$	0	0
$605$	2.62826	0.106854
$606$	0	0
$607$	36.8154	1.49429	0.747146	−	0.664660i	$-0.231423\pi$
$607$	36.8154	1.49429	0.747146	+	0.664660i	$0.231423\pi$
$608$	0	0
$609$	43.0328	1.74378
$610$	0	0
$611$	34.6831	1.40313
$612$	0	0
$613$	29.2501	1.18140	0.590700	−	0.806891i	$-0.298852\pi$
$613$	29.2501	1.18140	0.590700	+	0.806891i	$0.298852\pi$
$614$	0	0
$615$	5.86838	0.236636
$616$	0	0
$617$	−24.6495	−0.992350	−0.496175	−	0.868223i	$-0.665263\pi$
$617$	−24.6495	−0.992350	−0.496175	+	0.868223i	$0.665263\pi$
$618$	0	0
$619$	5.25010	0.211019	0.105510	−	0.994418i	$-0.466353\pi$
$619$	5.25010	0.211019	0.105510	+	0.994418i	$0.466353\pi$
$620$	0	0
$621$	−1.85471	−0.0744271
$622$	0	0
$623$	−4.01227	−0.160748
$624$	0	0
$625$	24.0861	0.963444
$626$	0	0
$627$	3.83931	0.153327
$628$	0	0
$629$	14.9255	0.595118
$630$	0	0
$631$	−16.5702	−0.659650	−0.329825	−	0.944042i	$-0.606990\pi$
$631$	−16.5702	−0.659650	−0.329825	+	0.944042i	$0.606990\pi$
$632$	0	0
$633$	−7.84819	−0.311938
$634$	0	0
$635$	1.48742	0.0590263
$636$	0	0
$637$	5.96497	0.236341
$638$	0	0
$639$	12.9882	0.513805
$640$	0	0
$641$	−3.22480	−0.127372	−0.0636859	−	0.997970i	$-0.520286\pi$
$641$	−3.22480	−0.127372	−0.0636859	+	0.997970i	$0.520286\pi$
$642$	0	0
$643$	37.5343	1.48021	0.740104	−	0.672493i	$-0.234776\pi$
$643$	37.5343	1.48021	0.740104	+	0.672493i	$0.234776\pi$
$644$	0	0
$645$	−0.454699	−0.0179038
$646$	0	0
$647$	−7.41628	−0.291564	−0.145782	−	0.989317i	$-0.546570\pi$
$647$	−7.41628	−0.291564	−0.145782	+	0.989317i	$0.546570\pi$
$648$	0	0
$649$	1.91070	0.0750014
$650$	0	0
$651$	55.0372	2.15708
$652$	0	0
$653$	30.1314	1.17913	0.589566	−	0.807720i	$-0.299299\pi$
$653$	30.1314	1.17913	0.589566	+	0.807720i	$0.299299\pi$
$654$	0	0
$655$	−1.12582	−0.0439896
$656$	0	0
$657$	10.3191	0.402586
$658$	0	0
$659$	0.607522	0.0236657	0.0118328	−	0.999930i	$-0.496233\pi$
$659$	0.607522	0.0236657	0.0118328	+	0.999930i	$0.496233\pi$
$660$	0	0
$661$	24.5944	0.956609	0.478305	−	0.878194i	$-0.341251\pi$
$661$	24.5944	0.956609	0.478305	+	0.878194i	$0.341251\pi$
$662$	0	0
$663$	−20.4745	−0.795165
$664$	0	0
$665$	−1.47831	−0.0573262
$666$	0	0
$667$	50.3461	1.94941
$668$	0	0
$669$	−12.2058	−0.471903
$670$	0	0
$671$	−3.37971	−0.130472
$672$	0	0
$673$	15.7523	0.607206	0.303603	−	0.952799i	$-0.401810\pi$
$673$	15.7523	0.607206	0.303603	+	0.952799i	$0.401810\pi$
$674$	0	0
$675$	1.39811	0.0538135
$676$	0	0
$677$	−20.0791	−0.771701	−0.385851	−	0.922561i	$-0.626092\pi$
$677$	−20.0791	−0.771701	−0.385851	+	0.922561i	$0.626092\pi$
$678$	0	0
$679$	−43.5127	−1.66987
$680$	0	0
$681$	−4.60382	−0.176419
$682$	0	0
$683$	−28.8494	−1.10389	−0.551946	−	0.833880i	$-0.686115\pi$
$683$	−28.8494	−1.10389	−0.551946	+	0.833880i	$0.686115\pi$
$684$	0	0
$685$	−2.74945	−0.105051
$686$	0	0
$687$	32.2865	1.23181
$688$	0	0
$689$	11.3926	0.434023
$690$	0	0
$691$	−25.9046	−0.985459	−0.492729	−	0.870183i	$-0.664001\pi$
$691$	−25.9046	−0.985459	−0.492729	+	0.870183i	$0.664001\pi$
$692$	0	0
$693$	4.07026	0.154617
$694$	0	0
$695$	2.16447	0.0821031
$696$	0	0
$697$	−23.1108	−0.875384
$698$	0	0
$699$	−24.0964	−0.911408
$700$	0	0
$701$	38.5710	1.45681	0.728403	−	0.685149i	$-0.240263\pi$
$701$	38.5710	1.45681	0.728403	+	0.685149i	$0.240263\pi$
$702$	0	0
$703$	16.3538	0.616794
$704$	0	0
$705$	5.82396	0.219343
$706$	0	0
$707$	41.3609	1.55554
$708$	0	0
$709$	17.3843	0.652880	0.326440	−	0.945218i	$-0.394151\pi$
$709$	17.3843	0.652880	0.326440	+	0.945218i	$0.394151\pi$
$710$	0	0
$711$	−10.8149	−0.405590
$712$	0	0
$713$	64.3906	2.41145
$714$	0	0
$715$	0.540319	0.0202068
$716$	0	0
$717$	−18.8866	−0.705335
$718$	0	0
$719$	36.9751	1.37894	0.689470	−	0.724314i	$-0.257844\pi$
$719$	36.9751	1.37894	0.689470	+	0.724314i	$0.257844\pi$
$720$	0	0
$721$	−42.9855	−1.60087
$722$	0	0
$723$	−43.9650	−1.63508
$724$	0	0
$725$	−37.9517	−1.40949
$726$	0	0
$727$	−30.9002	−1.14602	−0.573012	−	0.819547i	$-0.694225\pi$
$727$	−30.9002	−1.14602	−0.573012	+	0.819547i	$0.694225\pi$
$728$	0	0
$729$	−24.8599	−0.920739
$730$	0	0
$731$	1.79069	0.0662312
$732$	0	0
$733$	26.8696	0.992450	0.496225	−	0.868194i	$-0.334719\pi$
$733$	26.8696	0.992450	0.496225	+	0.868194i	$0.334719\pi$
$734$	0	0
$735$	1.00163	0.0369458
$736$	0	0
$737$	2.40011	0.0884093
$738$	0	0
$739$	−8.83782	−0.325105	−0.162552	−	0.986700i	$-0.551973\pi$
$739$	−8.83782	−0.325105	−0.162552	+	0.986700i	$0.551973\pi$
$740$	0	0
$741$	−22.4338	−0.824127
$742$	0	0
$743$	−1.74292	−0.0639414	−0.0319707	−	0.999489i	$-0.510178\pi$
$743$	−1.74292	−0.0639414	−0.0319707	+	0.999489i	$0.510178\pi$
$744$	0	0
$745$	3.02364	0.110778
$746$	0	0
$747$	−25.4466	−0.931043
$748$	0	0
$749$	−18.0620	−0.659970
$750$	0	0
$751$	−31.0485	−1.13298	−0.566489	−	0.824070i	$-0.691698\pi$
$751$	−31.0485	−1.13298	−0.566489	+	0.824070i	$0.691698\pi$
$752$	0	0
$753$	−9.44022	−0.344021
$754$	0	0
$755$	0.403054	0.0146686
$756$	0	0
$757$	39.6825	1.44228	0.721142	−	0.692787i	$-0.243617\pi$
$757$	39.6825	1.44228	0.721142	+	0.692787i	$0.243617\pi$
$758$	0	0
$759$	9.71674	0.352695
$760$	0	0
$761$	45.2359	1.63980	0.819900	−	0.572506i	$-0.194029\pi$
$761$	45.2359	1.63980	0.819900	+	0.572506i	$0.194029\pi$
$762$	0	0
$763$	34.5683	1.25146
$764$	0	0
$765$	−1.68493	−0.0609188
$766$	0	0
$767$	−11.1646	−0.403129
$768$	0	0
$769$	14.0058	0.505063	0.252531	−	0.967589i	$-0.418737\pi$
$769$	14.0058	0.505063	0.252531	+	0.967589i	$0.418737\pi$
$770$	0	0
$771$	44.0591	1.58675
$772$	0	0
$773$	−17.1013	−0.615089	−0.307545	−	0.951534i	$-0.599507\pi$
$773$	−17.1013	−0.615089	−0.307545	+	0.951534i	$0.599507\pi$
$774$	0	0
$775$	−48.5388	−1.74356
$776$	0	0
$777$	35.3769	1.26914
$778$	0	0
$779$	−25.3224	−0.907268
$780$	0	0
$781$	2.75431	0.0985571
$782$	0	0
$783$	−2.17534	−0.0777404
$784$	0	0
$785$	4.87093	0.173851
$786$	0	0
$787$	−13.1011	−0.467004	−0.233502	−	0.972356i	$-0.575019\pi$
$787$	−13.1011	−0.467004	−0.233502	+	0.972356i	$0.575019\pi$
$788$	0	0
$789$	32.0724	1.14181
$790$	0	0
$791$	34.9174	1.24152
$792$	0	0
$793$	19.7483	0.701283
$794$	0	0
$795$	1.91303	0.0678483
$796$	0	0
$797$	−4.29310	−0.152069	−0.0760347	−	0.997105i	$-0.524226\pi$
$797$	−4.29310	−0.152069	−0.0760347	+	0.997105i	$0.524226\pi$
$798$	0	0
$799$	−22.9359	−0.811413
$800$	0	0
$801$	−5.01068	−0.177044
$802$	0	0
$803$	2.18830	0.0772233
$804$	0	0
$805$	−3.74138	−0.131866
$806$	0	0
$807$	6.05979	0.213315
$808$	0	0
$809$	29.3499	1.03189	0.515944	−	0.856623i	$-0.327441\pi$
$809$	29.3499	1.03189	0.515944	+	0.856623i	$0.327441\pi$
$810$	0	0
$811$	−1.05794	−0.0371494	−0.0185747	−	0.999827i	$-0.505913\pi$
$811$	−1.05794	−0.0371494	−0.0185747	+	0.999827i	$0.505913\pi$
$812$	0	0
$813$	−37.5874	−1.31825
$814$	0	0
$815$	5.20935	0.182476
$816$	0	0
$817$	1.96205	0.0686435
$818$	0	0
$819$	−23.7833	−0.831057
$820$	0	0
$821$	35.9777	1.25563	0.627816	−	0.778362i	$-0.283949\pi$
$821$	35.9777	1.25563	0.627816	+	0.778362i	$0.283949\pi$
$822$	0	0
$823$	20.0377	0.698471	0.349236	−	0.937035i	$-0.386441\pi$
$823$	20.0377	0.698471	0.349236	+	0.937035i	$0.386441\pi$
$824$	0	0
$825$	−7.32465	−0.255012
$826$	0	0
$827$	−21.8395	−0.759432	−0.379716	−	0.925103i	$-0.623978\pi$
$827$	−21.8395	−0.759432	−0.379716	+	0.925103i	$0.623978\pi$
$828$	0	0
$829$	24.6610	0.856511	0.428256	−	0.903658i	$-0.359128\pi$
$829$	24.6610	0.856511	0.428256	+	0.903658i	$0.359128\pi$
$830$	0	0
$831$	20.4392	0.709029
$832$	0	0
$833$	−3.94462	−0.136673
$834$	0	0
$835$	0.247339	0.00855953
$836$	0	0
$837$	−2.78218	−0.0961661
$838$	0	0
$839$	−5.34359	−0.184481	−0.0922405	−	0.995737i	$-0.529403\pi$
$839$	−5.34359	−0.184481	−0.0922405	+	0.995737i	$0.529403\pi$
$840$	0	0
$841$	30.0495	1.03619
$842$	0	0
$843$	−12.2710	−0.422634
$844$	0	0
$845$	0.0582269	0.00200307
$846$	0	0
$847$	−24.5333	−0.842976
$848$	0	0
$849$	47.2737	1.62243
$850$	0	0
$851$	41.3891	1.41880
$852$	0	0
$853$	27.0621	0.926588	0.463294	−	0.886205i	$-0.346667\pi$
$853$	27.0621	0.926588	0.463294	+	0.886205i	$0.346667\pi$
$854$	0	0
$855$	−1.84617	−0.0631376
$856$	0	0
$857$	−16.6374	−0.568321	−0.284161	−	0.958777i	$-0.591715\pi$
$857$	−16.6374	−0.568321	−0.284161	+	0.958777i	$0.591715\pi$
$858$	0	0
$859$	−49.9417	−1.70399	−0.851995	−	0.523550i	$-0.824607\pi$
$859$	−49.9417	−1.70399	−0.851995	+	0.523550i	$0.824607\pi$
$860$	0	0
$861$	−54.7779	−1.86683
$862$	0	0
$863$	45.2817	1.54141	0.770703	−	0.637194i	$-0.219905\pi$
$863$	45.2817	1.54141	0.770703	+	0.637194i	$0.219905\pi$
$864$	0	0
$865$	1.76686	0.0600750
$866$	0	0
$867$	−27.6946	−0.940557
$868$	0	0
$869$	−2.29344	−0.0777996
$870$	0	0
$871$	−14.0243	−0.475196
$872$	0	0
$873$	−54.3405	−1.83915
$874$	0	0
$875$	5.67557	0.191869
$876$	0	0
$877$	−12.3342	−0.416496	−0.208248	−	0.978076i	$-0.566776\pi$
$877$	−12.3342	−0.416496	−0.208248	+	0.978076i	$0.566776\pi$
$878$	0	0
$879$	−36.8062	−1.24144
$880$	0	0
$881$	55.7846	1.87943	0.939715	−	0.341959i	$-0.111090\pi$
$881$	55.7846	1.87943	0.939715	+	0.341959i	$0.111090\pi$
$882$	0	0
$883$	49.1991	1.65568	0.827841	−	0.560963i	$-0.189569\pi$
$883$	49.1991	1.65568	0.827841	+	0.560963i	$0.189569\pi$
$884$	0	0
$885$	−1.87474	−0.0630188
$886$	0	0
$887$	−27.4379	−0.921274	−0.460637	−	0.887589i	$-0.652379\pi$
$887$	−27.4379	−0.921274	−0.460637	+	0.887589i	$0.652379\pi$
$888$	0	0
$889$	−13.8842	−0.465660
$890$	0	0
$891$	−5.70871	−0.191249
$892$	0	0
$893$	−25.1307	−0.840967
$894$	0	0
$895$	−0.950913	−0.0317855
$896$	0	0
$897$	−56.7768	−1.89572
$898$	0	0
$899$	75.5220	2.51880
$900$	0	0
$901$	−7.53389	−0.250990
$902$	0	0
$903$	4.24436	0.141243
$904$	0	0
$905$	−2.43958	−0.0810945
$906$	0	0
$907$	1.76246	0.0585214	0.0292607	−	0.999572i	$-0.490685\pi$
$907$	1.76246	0.0585214	0.0292607	+	0.999572i	$0.490685\pi$
$908$	0	0
$909$	51.6533	1.71323
$910$	0	0
$911$	−44.9273	−1.48851	−0.744254	−	0.667897i	$-0.767195\pi$
$911$	−44.9273	−1.48851	−0.744254	+	0.667897i	$0.767195\pi$
$912$	0	0
$913$	−5.39628	−0.178591
$914$	0	0
$915$	3.31612	0.109627
$916$	0	0
$917$	10.5089	0.347035
$918$	0	0
$919$	−33.3449	−1.09995	−0.549973	−	0.835182i	$-0.685362\pi$
$919$	−33.3449	−1.09995	−0.549973	+	0.835182i	$0.685362\pi$
$920$	0	0
$921$	−32.7112	−1.07787
$922$	0	0
$923$	−16.0940	−0.529740
$924$	0	0
$925$	−31.1998	−1.02584
$926$	0	0
$927$	−53.6821	−1.76315
$928$	0	0
$929$	−25.9145	−0.850226	−0.425113	−	0.905140i	$-0.639766\pi$
$929$	−25.9145	−0.850226	−0.425113	+	0.905140i	$0.639766\pi$
$930$	0	0
$931$	−4.32210	−0.141651
$932$	0	0
$933$	78.4864	2.56953
$934$	0	0
$935$	−0.357311	−0.0116853
$936$	0	0
$937$	−44.6469	−1.45855	−0.729275	−	0.684221i	$-0.760142\pi$
$937$	−44.6469	−1.45855	−0.729275	+	0.684221i	$0.760142\pi$
$938$	0	0
$939$	17.5015	0.571141
$940$	0	0
$941$	41.4230	1.35035	0.675175	−	0.737658i	$-0.264068\pi$
$941$	41.4230	1.35035	0.675175	+	0.737658i	$0.264068\pi$
$942$	0	0
$943$	−64.0873	−2.08697
$944$	0	0
$945$	0.161657	0.00525869
$946$	0	0
$947$	−20.3539	−0.661413	−0.330707	−	0.943734i	$-0.607287\pi$
$947$	−20.3539	−0.661413	−0.330707	+	0.943734i	$0.607287\pi$
$948$	0	0
$949$	−12.7866	−0.415072
$950$	0	0
$951$	−31.2960	−1.01484
$952$	0	0
$953$	−45.8031	−1.48371	−0.741854	−	0.670562i	$-0.766053\pi$
$953$	−45.8031	−1.48371	−0.741854	+	0.670562i	$0.766053\pi$
$954$	0	0
$955$	0.425369	0.0137646
$956$	0	0
$957$	11.3965	0.368396
$958$	0	0
$959$	25.6645	0.828751
$960$	0	0
$961$	65.5897	2.11580
$962$	0	0
$963$	−22.5565	−0.726874
$964$	0	0
$965$	−5.68683	−0.183066
$966$	0	0
$967$	−22.5611	−0.725515	−0.362757	−	0.931884i	$-0.618165\pi$
$967$	−22.5611	−0.725515	−0.362757	+	0.931884i	$0.618165\pi$
$968$	0	0
$969$	14.8354	0.476583
$970$	0	0
$971$	−4.81645	−0.154567	−0.0772836	−	0.997009i	$-0.524625\pi$
$971$	−4.81645	−0.154567	−0.0772836	+	0.997009i	$0.524625\pi$
$972$	0	0
$973$	−20.2041	−0.647714
$974$	0	0
$975$	42.7993	1.37068
$976$	0	0
$977$	−18.0275	−0.576751	−0.288376	−	0.957517i	$-0.593115\pi$
$977$	−18.0275	−0.576751	−0.288376	+	0.957517i	$0.593115\pi$
$978$	0	0
$979$	−1.06258	−0.0339602
$980$	0	0
$981$	43.1703	1.37832
$982$	0	0
$983$	0.0152001	0.000484808 0	0.000242404	−	1.00000i	$-0.499923\pi$
$983$	0.0152001	0.000484808 0	0.000242404		1.00000i	$0.499923\pi$
$984$	0	0
$985$	−5.44343	−0.173442
$986$	0	0
$987$	−54.3633	−1.73040
$988$	0	0
$989$	4.96567	0.157899
$990$	0	0
$991$	−34.5992	−1.09908	−0.549540	−	0.835467i	$-0.685197\pi$
$991$	−34.5992	−1.09908	−0.549540	+	0.835467i	$0.685197\pi$
$992$	0	0
$993$	−27.6502	−0.877452
$994$	0	0
$995$	2.00601	0.0635949
$996$	0	0
$997$	28.0146	0.887232	0.443616	−	0.896217i	$-0.353695\pi$
$997$	28.0146	0.887232	0.443616	+	0.896217i	$0.353695\pi$
$998$	0	0
$999$	−1.78833	−0.0565803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.d.1.10	✓	12
4.3	odd	2		2672.2.a.p.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.d.1.10	✓	12	1.1	even	1	trivial
2672.2.a.p.1.3		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q+2.42555 q^{3} -0.247339 q^{5} +2.30877 q^{7} +2.88329 q^{9} +O(q^{10})\)	\(q+2.42555 q^{3} -0.247339 q^{5} +2.30877 q^{7} +2.88329 q^{9} +0.611439 q^{11} -3.57276 q^{13} -0.599934 q^{15} +2.36266 q^{17} +2.58875 q^{19} +5.60004 q^{21} +6.55175 q^{23} -4.93882 q^{25} -0.283087 q^{27} +7.68437 q^{29} +9.82801 q^{31} +1.48308 q^{33} -0.571050 q^{35} +6.31725 q^{37} -8.66590 q^{39} -9.78170 q^{41} +0.757915 q^{43} -0.713151 q^{45} -9.70767 q^{47} -1.66957 q^{49} +5.73074 q^{51} -3.18874 q^{53} -0.151233 q^{55} +6.27914 q^{57} +3.12492 q^{59} -5.52747 q^{61} +6.65686 q^{63} +0.883683 q^{65} +3.92535 q^{67} +15.8916 q^{69} +4.50464 q^{71} +3.57893 q^{73} -11.9794 q^{75} +1.41167 q^{77} -3.75089 q^{79} -9.33651 q^{81} -8.82555 q^{83} -0.584378 q^{85} +18.6388 q^{87} -1.73784 q^{89} -8.24868 q^{91} +23.8383 q^{93} -0.640299 q^{95} -18.8467 q^{97} +1.76296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * q^99

Introduction

L-functions

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